Rotational Symmetry

Rotational Symmetry occurs if an object can be rotated less than \(360^\circ\) and be unchanged. The center of the rotation is called the point of symmetry.

Take for example some of the letters of the alphabet, which have rotational symmetry as they can be rotated \(180^\circ\) and remain the same.

Problem Solving

If point \((81,62)\) is rotated about point \((25,64)\) by \(30^{\circ}\) in each rotation, how many rotations will be required to get the final point again as \((81,62)\)?

We have to rotate the point by \(360^{\circ}\) to get it back. So the rotations required are \(\frac {360^{\circ}}{30^{\circ}}\)= \(12\) rotations

Let \(f (x)\) be the relation from \(N->\){\(5,10\)} equal to \(10^{\circ}\) when \(x\) is odd and \(5^{\circ}\) when \(x\) is even. Here \(x\) denotes \(x^\text{th}\) rotation of point \((2,5)\) and \(f (x)\) is angle rotated clockwise by this \(x^\text{th}\) rotation when rotated about the point \((8,15)\). If after \(a\) rotations, the final point is again \((2,5)\), find the minimum value of \(\frac {a}{2}\).